A103916 - OEIS

%I #13 Jul 21 2019 09:03:24

%S 1,14,439,24940,2250621,296266266,53624576979,12780684581400,

%T 3880806293223225,1462807581365269350,670261417348408188975,

%U 366936357918296751120900,236559234981486279096163125

%N Column k=2 sequence (without zero entries) of table A060524.

%C a(n) = sum over all multinomials M2(2*(n+1),k), k from {1..p(2*(n+1))} restricted to partitions with exactly two odd and any nonnegative number even parts. p(2*(n+1)) = A000041(2*(n+1)) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*(n+1),k). - _Wolfdieter Lang_, Aug 07 2007

%F E.g.f. (with alternating zeros): A(x) = (d^2/dx^2)a(x) with a(x):=(1/(sqrt(1-x^2))*(log(sqrt((1+x)/(1-x))))^2)/2!.

%F a(n) ~ log(2*n)^2 * 2^(2*n) * n^(2*n + 2) / (exp(2*n)) * (1 + (2*gamma + 6*log(2))/log(2*n) + (gamma^2 + 6*gamma*log(2) + 9*log(2)^2 - Pi^2/2) / log(2*n)^2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jul 21 2019

%e Multinomial representation for a(2): partitions of 2*3=6 with two odd parts: (1,5) with A-St position k=2; (3^2) with k=4; (1^2,4) with k=5; (1,2,3) with k=6 and (1^2,2^2) with k=9. The M2 numbers for these partitions are 144, 40, 90, 120, 45, adding up to 439 = a(2).

%t nmax = 20; Table[(CoefficientList[Series[(4 + 8*x*Log[(1 + x)/(1 - x)] + (1/2 + x^2)*Log[(1 + x)/(1 - x)]^2)/(4*(1 - x^2)^(5/2)), {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[2*n + 1]], {n, 0, nmax}] (* _Vaclav Kotesovec_, Jul 21 2019 *)

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Feb 24 2005